Condition (\(\mathcal {C}\) .REF ) contains the main ingredients to derive sparsity properties, consistency and asymptotic normality. We first note that if \(\lambda _{n,j}=\lambda _n\) , then \(a_n=b_n=\lambda _n\) , whereby it is easily deduced that the two conditions on \(a_n\)  and \(b_n\)  cannot be satisfied simultaneously even if \(p_n=p\) . This justifies the introduction of an adaptive version of the Dantzig selector and motivates the use of the adaptive lasso. The condition \(a_n\sqrt{s_n \mu _n}\rightarrow 0\)  for the adaptive Lasso is similar to the one imposed by [1]} when \(\mu _n\)  is replaced by \(n\)  and \(s_n=s\)  in their context. However, we require a slightly stronger condition on \(b_n\)  than the one required by [1]}. In our setting, their assumption would be written as \(b_n \sqrt{\mu _n/p_n} \rightarrow \infty \) . However, we would have to assume that \(\nu _{\max }\big (\mathbf {A}_{n}(\beta _0)\big )=O(\mu _n)\) . Such a condition is not straightforwardly satisfied in our setting since, for instance, the conditions (\(\mathcal {C}\) .REF )-(\(\mathcal {C}\) .REF ) only imply that \(\nu _{\max }\big (\mathbf {A}_{n}(\beta _0)\big )=O({p_n \mu _n})\) .
